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Arnold's Trivium
1991 Arnold's Trivium
66
Problem 66
Problem 66
Source:
July 1, 2010
logarithms
complex analysis
complex numbers
Problem Statement
Solve the Dirichlet problem
Δ
u
=
0
for
x
2
+
y
2
<
1
\Delta u=0\text{ for }x^2+y^2<1
Δ
u
=
0
for
x
2
+
y
2
<
1
u
=
1
for
x
2
+
y
2
=
1
,
y
>
0
u=1\text{ for }x^2+y^2=1,\;y>0
u
=
1
for
x
2
+
y
2
=
1
,
y
>
0
u
=
−
1
for
x
2
+
y
2
=
1
,
y
<
0
u=-1\text{ for }x^2+y^2=1,\;y<0
u
=
−
1
for
x
2
+
y
2
=
1
,
y
<
0
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